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Transient and busy period analysis of the GIG/1 Queue as a Hilbert factorization problem

Published online by Cambridge University Press:  14 July 2016

Dimitris J. Bertsimas ,
Julian Keilson ,
Daisuke Nakazato  and
Hongtao Zhang
Dimitris J. Bertsimas*
Affiliation:
Sloan School of Management, MIT
Julian Keilson*
Affiliation:
Sloan School of Management, MIT
Daisuke Nakazato*
Affiliation:
Operations Research Center, MIT
Hongtao Zhang*
Affiliation:
Operations Research Center, MIT
*
∗ Postal address: Sloan School of Management
∗ Postal address: Sloan School of Management
∗∗ Operations Research Center, MIT, Cambridge, MA 02139, USA.
∗∗ Operations Research Center, MIT, Cambridge, MA 02139, USA.
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Abstract

In this paper we find the waiting time distribution in the transient domain and the busy period distribution of the GI G/1 queue. We formulate the problem as a two-dimensional Lindley process and then transform it to a Hilbert factorization problem. We achieve the solution of the factorization problem for the GI/R/1, R/G/1 queues, where R is the class of distributions with rational Laplace transforms. We obtain simple closed-form expressions for the Laplace transforms of the waiting time distribution and the busy period distribution. Furthermore, we find closed-form formulae for the first two moments of the distributions involved.

Keywords

TRANSIENT ANALYSISBUSY PERIODLINDLEY EQUATION
Type
Research Papers
Information
Journal of Applied Probability , Volume 28 , Issue 4 , December 1991 , pp. 873 - 885
Copyright
Copyright © Applied Probability Trust 1991 

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Footnotes

The research of Dr Bertsimas was partially supported by grants from the Leaders for Manufacturing program at MIT and from Draper Laboratory.

References

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